new features of extended wormhole solutions in the scalar field gravity theories
TRANSCRIPT
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NEW FEATURES OF EXTENDED WORMHOLE SOLUTIONS
IN THE SCALAR FIELD GRAVITY THEORIES
Kamal K. Nandi1,2,a, Ilnur Nigmatzyanov2,b, Ramil Izmailov2,c and Nail G.Migranov2,d
1Department of Mathematics, University of North Bengal, Siliguri (W.B.) 734013, India
2Joint Research Laboratory, Bashkir State Pedagogical University, 3-A,October Revolution Str., Ufa 450000, Russia
aEmail: [email protected]: [email protected]
cEmail:[email protected]: [email protected]
Abstract
The present paper reports interesting new features that wormhole solutionsin the scalar field gravity theory have. To demonstrate these, we obtain, by us-ing a slightly modified form of the Matos-Nunez algorithm, an extended classof asymptotically flat wormhole solutions belonging to Einstein minimally cou-pled scalar field theory. Generally, solutions in these theories do not representtraversable wormholes due to the occurrence of curvature singularities. How-ever, the Ellis I solution of the Einstein minimally coupled theory, when Wickrotated, yields Ellis class III solution, the latter representing a singularity-freetraversable wormhole. We see that Ellis I and III are not essentially independentsolutions. The Wick rotated seed solutions, extended by the algorithm, containtwo new parameters a and δ. The effect of the parameter a on the geodesic mo-tion of test particles reveals some remarkable features. By arguing for Sagnaceffect in the extended Wick rotated solution, we find that the parameter a canindeed be interpreted as a rotation parameter of the wormhole. The analysesreported here have wider applicability in that they can very well be adopted inother theories, including in the string theory.
I. Introduction
Recently, there is a revival of interest in the scalar field gravity theories in-cluding the Brans-Dicke Theory due principally to the following reasons: Suchtheories occur naturally in the low energy limit of the effective string theoryin four dimensions or the Kaluza-Klein theory. It is found to be consistentnot only with the weak field solar system tests but also with the recent cos-mological observations. Moreover, the theory accommodates Mach’s principle.(It is known that Einstein’s General Relativity can not accommodate Mach’sprinciple satisfactorily). All these information are well known.
1
A less well known yet an important arena where Brans-Dicke Theory hasfound immense applications is the field of wormhole physics, a field recentlyre-activated by the seminal work of Morris, Thorne and Yurtsever [1]. Concep-tual predecessors of modern day’s wormholes could be traced to the geometry ofFlamm paraboloid, Wheeler’s concept of “charge without charge”, Klein bottleor the Einstein-Rosen bridge model of a charged particle [2]. Wormholes aretopological handles that connect two distant regions of space. These objectsare invoked in the investigations of problems ranging from local to cosmologicalscales, not to mention the possibility of using these objects as a means of inter-stellar travel [1]. Wormholes require for their construction what is called “exoticmatter” - matter that violates some or all of the known energy conditions, theweakest being the averaged null energy condition. Such matters are known toarise in quantum effects (Casimir effect, for example). However, the strongesttheoretical justification for the existence of exotic matter comes from the notionof dark energy or phantom energy that are necessary to explain the present ac-celeration of the Universe. Some classical fields can be conceived to play the roleof exotic matter. They are known to occur, for instance, in the R+R2 theories[3], Visser’s thin shell geometries [4] and, of course, in scalar-tensor theories [5]of which Brans-Dicke theory is a prototype. There are several other situationswhere the energy conditions could be violated [6].
Brans-Dicke theory describes gravitation through a metric tensor ( gµν) anda massless scalar field ( φ). The Brans-Dicke action for the coupling parameterω = −1 can be obtained in the Jordan frame from the vacuum linear stringtheory in the low energy limit. The action can be conformally rescaled intowhat is known as the Einstein frame action in which the scalar field couplesminimally to gravity. The last is referred to as the Einstein minimally coupledscalar field theory. Several static (mostly nontraversable) wormhole solutionsin Einstein minimally coupled scalar field theory and Brans-Dicke theory havebeen widely investigated in the literature [7]. However, to our knowledge, exactrotating wormhole solutions are relatively scarce except a recent one in Einsteinminimally coupled scalar field theory discussed by Matos and Nunez [8]. In thiscontext, we recall the well known fact that the formal independent solutionsof Brans-Dicke theory are not unique. (Of course, the black hole solution isunique for which the Brans-Dicke or minimal scalar field is trivial in virtueof the so called “no scalar hair” theorem.) Four classes of static Brans-Dicketheory solutions were derived by Brans [9] himself way back in 1962, and thecorresponding four classes of Einstein minimally coupled field theory solutionsare also known [10]. But recently it has been shown that only two of the fourclasses of Brans’ solutions are independent [11]; the other two can be derivedfrom them. However, although all the original four classes of Brans’ or Einsteinminimally coupled solutions are important in their own right, we shall hereconsider, for illustrative purposes, only one of them (Ellis I) as seed solution.The same procedure can be easily adopted in other three classes.
The general motivation in the present paper is the following: To properlyframe an algorithm for generating singularity-free asymptotically flat rotatingwormhole solutions from the Ellis seed solutions and to investigate the role of
2
new parameters in the extended solutions. The analyses also answer a certainlong standing query on the wormhole solutions in the Brans-Dicke theory.
In this article, using a slightly modified algorithm of Matos and Nunez [8],we shall provide a method for generating wormhole solutions from the knownstatic seed solutions belonging to Einstein minimally coupled scalar field theory.The solutions can then be transferred to those of Brans-Dicke theory via inverseDicke transformations. For illustration of the method, only Ellis I seed solutionis considered here, others are left out because they can be dealt with similarly.The Brans-Dicke solutions can be further rephrased as solutions of the vacuum4-dimensional low energy string theory ( ω = −1) and the section II shows howto do that. In sections II-V, we shall analyze and compare the behavior Ellis IIIand the Wick rotated Ellis I solution pointing out certain interesting differencesin these geometries. In section VI, the study of the geodesic motion in theextended Wick rotated Ellis I solution reveals the meanings of the Matos-Nunezparameter a and section VII shows, via consideration of the Sagnac effect, thata can indeed be accepted as a rotation parameter. Finally, in section VIII,we shall summarize the results. Throughout the article, we take the signature(−,+,+,+) and units such that 8πG = c = 1, unless restored specifically. Greekindices run from 0 to 3 while Roman indices run from 1 to 3.
II. The action, ansatz and the algorithm
Let us start from the 4-dimensional, low energy effective action of heteroticstring theory compactified on a 6-torus. The tree level string action, keepingonly linear terms in the string tension α′ and in the curvature R, takes thefollowing form in the matter free region ( Smatter = 0):
Sstring =1
α′
∫d4x√−ge−2eΦ
[R + 4gµνΦ,µΦ,ν
], (1)
where gµν is the string metric and Φ is the dilaton field. Note that the zerovalues of other matter fields do not impose any additional constraints either on
the metric or on the dilation [12]. Under the substitution e−2eΦ = φ, the aboveaction reduces to the BD action
SBD =
∫d4x√−g[φR +
1
φgµνφ,µφ,ν
], (2)
in which the BD coupling parameter ω = −1. This particular value is actuallymodel independent and it actually arises due to the target space duality. Itshould be noted that the BD action has a conformal invariance characterizedby a constant gauge parameter ξ [13]. Arbitrary values of ξ can actually leadto a shift from the value ω = −1, but we fix this ambiguity by choosing ξ = 0.Under a further substitution
gµν = φgµν
dϕ =
√2ω + 3
2α
dφ
φ;α 6= 0;ω 6= 3
2, (3)
3
in which we have introduced, on purpose, a constant parameter α that can haveany sign. Then the action (2) goes into the form of EMS action
SEMS =
∫d4x
√−g[R+ αgµνϕ,µϕ,ν
]. (4)
The EMS field equations are given by
Rµν = −αϕ,µϕ,ν (5)
ϕ:µ;µ = 0 (6)
We shall choose α = +1, ϕ = ϕ(l) in what follows. The negative sign on theright hand side of Eq.(5) implies that the source stresses violate some energyconditions. The ansatz we take is the following:
ds2 = −f(l) (dt+ a cos θdψ)2+ f−1(l)
[dl2 + (l2 + l20)
(dθ2 + sin2 θdψ2
)], (7)
where l0 is an arbitrary constant, the constant a has been interpreted in [8] as arotational parameter of the wormhole. We call it the Matos-Nunez parameter.The ansatz in (7) is actually a subclass of the more general class of stationarymetrics given by [14,15]:
ds2 = −f(dt− ωidx
i)2
+ f−1hijdxidxj (8)
where the metric function f , the vector potential ωi and the reduced metric hij
depend only on space coordinates xi. We shall see below that the parametera can be so adjusted as to make a symmetric, traversable wormhole out of anasymmetric, nontraversable one. Note also that the replacement of a → −adoes not alter the field equations.
The function f(l) of the ansatz (7) is then a solution of the field equations(5) and (6) if it satisfies the following:
[(l2 + l20
) f ′
f
]′+
a2f2
l2 + l20= 0, (9)
(f
′
f
)2
+4l20 + a2f2
(l2 + l20)2 − 2ϕ′2 = 0, (10)
where the prime denotes differentiation with respect to l.Algorithm:
Let f0 ≡ f0(l; p, q; a = 0) and ϕ0 = ϕ0(l; p, q; a = 0) be a known seed solutionset of the static configuration in which p, q are arbitrary constants in the solutioninterpreted as the mass and scalar charge of the configuration. Then the newgenerated (or extended) solution set (f, ϕ) is
f(l; p, q; a) =2npqδf−1
0
a2 + nδ2f−20
, ϕ(l; p, q; a) = ϕ0 (11)
4
where n is a natural number and the paremeters p, q are specific to a givenseed solution set (f0, ϕ0) while δ is a free parameter allowed by the generatedsolution in the sense that it cancels out of the nonlinear field equations. Thescalar field ϕ0 is remarkably given by the same static solution of the masslessKlein-Gordon equation ϕ:µ
;µ = 0. The seed solution ( a = 0) following fromEqs.(9) and (10) gives δ = 2pq. For the generated solution (a 6= 0), the value ofδ may be fixed either by the condition of asymptotic flatness or via the matchingconditions at specified boundaries. Eq.(11) is the algorithm we propose. Thisis similar to, but not quite the same as, the Matos-Nunez [8] algorithm. Thedifference is that they defined the free parameter as δ =
√D . The difficulty in
this case is that, for our seed solution set (f0, ϕ0) below, the field Eqs.(10) and(11) identically fix δ2 = D = 0 giving f = 0 which is obviously meaningless. Theother difference is that we have introduced a real number n that now designateseach seed solution f0 and likewise the corresponding new solution f . With theknown parameters n, p and q plugged into the right side of Eq.(11), the newsolution set (f, ϕ) identically satisfies Eqs. (9) and (10). One also sees that thealgorithm can be applied with the set (f, ϕ) as the new seed solution and theprocess can be indefinitely iterated to generate any number of new solutions.This is a notable generality of the algorithm.
III. Ellis I solution and its geometry
The study of the solutions of the Einstein minimally coupled scalar fieldsystem has a long history. Static spherically symmetric solutions have beenindependently discovered in different forms by many authors and their propertiesare well known [16,17]. We start from the following form of Class I solution,due to Buchdahl [18], of Einstein minimally coupled theory :
ds2 = −(
1 − m2r
1 + m2r
)2β
dt2 +(1 − m
2r
)2(1−β) (1 +
m
2r
)2(1+β)
×
[dr2 + r2dθ2 + r2 sin2 θdψ2] (12)
ϕ(r) =
√2(β2 − 1)
αln
[1 − m
2r
1 + m2r
], (13)
where m and β are two arbitrary constants. The same solution, in harmoniccoordinates, has been obtained and analyzed also by Bronnikov [19].
The metric (13) can be expanded to give
ds2 = −[1 − 2M
r+
2M2
r2+O
(1
r3
)]dt2 +
[1 +
2M
r+O
(1
r2
)]×
[dr2 + r2dθ2 + r2 sin2 θdψ2], (14)
from which one can read off the Keplerian mass
M = mβ (15)
5
The solution (12) describes all weak field solar system tests because it exactlyreproduces the PPN parameters. For β = 1, it reduces to the Schwarzschildblack hole solution in isotropic coordinates. For α = +1 and β > 1, it representsa naked singularity. The metric is invariant in form under inversion of the radial
coordinate r → m2
4r and we have two asymptotically flat regions (at r = 0 and
r = ∞), the minimum area radius (throat) occurring at r0 = m2
[β +
√β2 − 1
].
Thus, real throat is guaranteed by the condition β2 > 1 which we might call herethe wormhole condition. However, despite these facts, because of the occurrenceof naked singularity at r = m/2, it is not traversable and so Visser [4] called it
a “diseased” wormhole. For the choice α = +1, the quantity√
2(β2 − 1) is real
such that there is a real scalar charge σ from Eq.(14) given by
ϕ =σ
r= −2m
r
√β2 − 1
2(16)
But, in this case, we have violated almost all energy conditions importing byhand a negative sign before the kinetic term in Eq.(5). Alternatively, we couldhave chosen α = −1 in Eq.(13), which would then give an imaginary chargeσ. In both cases, however, we end up with the same equation Rµν = −ϕ,µϕ,ν .There is absolutely no problem in accommodating an imaginary scalar chargein any configuration violating energy conditions [17,18].
The Ricci scalar R for the solution (12) is
R =2m2r4(1 − β2)
(r −m/2)2(2−β)(r +m/2)2(2+β)(17)
which diverges at r = m/2 showing a curvature singularity there. For β ≥ 2, thedivergence in the Ricci scalar is removed, but then the metric becomes singular.However, metric singularity is often removable when one redefinines it in bettercoordinates and parameters.
Using the coordinate transformation l = r + m2
4r , the solution (12) and (13)can be expressed as
ds2 = −f0(l)dt2 +1
f0(l)
[dl2 + (l2 −m2)
(dθ2 + sin2 θdψ2
)],
f0(l) =
(l −m
l +m
)β
, (18)
ϕ0(l) =
√β2 − 1
2ln
[l −m
l +m
]. (19)
In this form, it is exactly the Ellis I solution that has been discussed also byBronnikov and Shikin [20]. Eqs.(18), (19) identically satisfy the field Eqs.(9)and (10) for a = 0. This is our seed solution set but we still need to suitablyredefine it because of the apprearance of the naked singularity at l = m The
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throat l0 appears at l0 = r0 + m2
4r0= mβ > m corresponding to r = r0. Thus
the minimum surface area now has a value 4πm2β2. For this solution, p = m,q = β, and it turns out that the seed solutions (18), (19) correspond to n = 4so that the generated EMS solution that identically satisfies the field equations(9) and (10) for a 6= 0 are:
f(l;m,β, a) =8mβδf−1
0
a2 + 4δ2f−20
;ϕ(l) =
√β2 − 1
2ln
[l −m
l +m
]. (20)
To achieve asymptotic flatness at both sides, that is, f(l) → 1 as l → ±∞, wenote that f0(l) → 1 as l → ±∞ . Therefore, we must fix
δ =2M ±
√4M2 − a2
2. (21)
In the above, we should retain only the positive sign before the square root. Thereason is that, for a = 0, the negative sign gives δ = 0 implying f = 0 which ismeaningless. On the other hand, the positive root gives δ = 2M and f = f0,as desired. Note that β = 1 does not lead to Kerr black hole solution from thegenerated metric. Therefore, the latter does not represent a rotating black holebut might represent the spacetime of a rotating wormhole [8].
Let us now examine wormhole geometries in the static and generated solu-tions in the EMS.
(a) Static seed case (a = 0):The first observation is that the metric functions in Eq.(18) diverge at the
singularity l = ±m as does the Ricci scalar. The next observation relates tothe behavior of the area radius. It exhibits certain peculiar properties for themetric in (18) for β > 1. For the segments l ≥ m and l ≤ −m, we have the
area radius ρI0(l) =
√f−10 (l2 −m2). Then, the area 4πρ2
0(l) decreases from +∞at one asymptotic flat end to a minimum value ρ0 min = ρ0(l0) at the throatl = l0 = mβ, and then becomes asymptotically large, but not flat, at a radialpoint l = m. In the remaining segment, we have |l| < m, and the area now has
to be redefined as ρII0 (l) =
√f−10 |m2 − l2|. It then decreases to zero at l = −m
(another throat, zero radius!) and then opens asymptotically out to −∞ at theother asymptotic flat end. Though in the r−coordinate version, the metric isinversion symmetric, there are now two asymptotically flat, isometric universeswith their own throats and are actually disconnected by the naked singularityat l = m. They are also asymmetric around l = 0 due to the fact that thethroat radii in the two universes are different.
(b) Generated case (a 6= 0):Here again, the metric functions given by Eq.(20) diverge at l = ±m. The
throat of the rotating wormhole can be found from the roots of the equation forl :
a2[l(f20 − 1) +mβ((f2
0 + 1)] + 4mβ(l −mβ)(2mβ +
√4m2β2 − a2) = 0 (22)
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They can be computed only numerically for given values of the parameters mand β. However, the area radius ρ(l) =
√f−1(l2 −m2) for the solution f shows
that, for a 6= 0, the area jumps to infinity for β > 1 at l = ±m but flaresout asymptotically to ±∞ on both sides. One now has three disconnected uni-verses, that is, a one-sided asymptotically flat universe, a both-sided non-flatbut asymptotically large “sandwich universe” and another one-sided asymptot-ically flat universe. For the extreme case a = 2mβ, the picture is the samebut the behavior of ρ(l) is symmetric around l = 0. Thus, the wormhole is nottraversable, be it seed solution (18) or extended solution (20). A natural ques-tion arises if they be made traversable by removing the singularity manifestedin the infinite jump in the area at l = ±m and in the curvature. We shallconsider this question now.
IV. Ellis III solution via Wick rotation
One procedure to remove the aforementioned singularities is to analyticallycontinue the Ellis I solutions (f0,ϕ0) by means of Wick rotation of the param-eters while maintaining the real numerical value of the throat radius. In thesolution set (f0,ϕ0), we choose
m→ −im, β → iβ (23)
so that l0 = mβ is invariant in sign and magnitude.(a) a = 0:Then the metric resulting from the seed Eq.(18) is and it is our redefined
seed solution:
ds2 = −f ′0(l)dt
2 +1
f ′0(l)
[dl2 + (l2 +m2)
(dθ2 + sin2 θdψ2
)](24)
f ′0(l) = exp
[−2β arccot
(l
m
)](25)
ϕ′0(l) =
[√2
√1 + β2
]arccot
(l
m
)(26)
This is no new solution but is just the Ellis III solution [19] which can beobtained in the original form by using the relation [20]
arccot(x) + arctan(x) = +π
2;x > 0 (27)
= −π2
;x < 0 (28)
and the function on the left shows a finite jump (of magnitude π) at x = 0.Thus, we get from Eqs.(25) and (26) two branches, the +ve sign corresponds tothe side l > 0 and the −ve sign to l < 0 [22]:
fEllis0± (l) = exp[−2β±π
2− arctan(
l
m)] (29)
8
ϕEllis0± (l) =
[√2
√1 + β2
](±π
2− arctan(
l
m)
)(30)
We might study the solutions (30) and (31) per se, or equivalently, study thetwo restricted branches taken together, while allowing for a discontinuity at theorigin l = 0. Otherwise, we might disregard (30), (31) and treat each of the± set in Eqs.(34), (35) as independently derived exact solution valid in the un-restricted range of l with no discontinuity at l = 0. The two alternatives donot appear quite the same. In fact, each of the individual branch representsa geodesically complete, asymptotically flat traversable wormhole (termed as”drainholes” by Ellis) having different masses, one positive and the other nega-tive, on two sides respectively. The known Ellis III solution is the +ve branchwhich is continuous over the entire interval l ∈ (−∞,+∞). The −ve branch isalso equally good. What we have shown here is that the Ellis solutions I andIII are not independent solutions of the Einstein minimally coupled scalar fieldtheory as one can be obtained from the other.
It is of interest to compare the behaviors of the Ellis III solutions (34)with the Wick rotated Ellis I solutions (30): (i) The Ellis III metric functionfEllis0+ (l) → 1 as l → +∞ but fEllis
0+ (l) → e−2πβ as l → −∞. These two limitscorrespond to a Schwarzschild mass M at one mouth and −Meπβ at the other.There is no discontinuity at the origin because fEllis
0+ (l) → e−πβ as l → ±0.In the solution (30), on the other hand, there is a discontinuity at the originbecause f0(l) → e±πβ as l → ±0 while there is no asymptotic mass jump sincef0(l) → 1 as l → ±∞. The curvature scalars for both (30) and (34) are formallythe same and given by
R′0 = −2m2(1 + β2)
(l2 +m2)2exp
[−2β arccot
(l
m
)](31)
REllis0+ = −2m2(1 + β2)
(l2 +m2)2exp
[−2βπ
2− arctan
(l
m
)]
(32)
which go to zero as l → ±∞. That means, the spacetime is flat on two sides forboth the solutions. Next, we verify what happens to these scalars at the singular
coordinate radius (r = m/2) that has now been shifted to the origin l = r−m2
4r =
0. (ii) The Ellis curvature scalar REllis0+ → − 2(1+β2)
m2 e−πβ as l → ±0 whereas the
curvature scalar R′0 exhibits a finite jump from − 2(1+β2)
m2 e−πβ to − 2(1+β2)m2 e+πβ
as l → ±0. (iii) The area radius ρEllis0+ (l) =
√f−1(Ellis)0+ (l2 +m2) → m
√eπβ as
l → ±0 whereas ρ′0(l) =√f ′−10 (l2 +m2) shows a finite jump from m
√eπβ to
m√e−πβ as l → ±0. These show that while Ellis wormhole (34) is traversable,
but jumps at the origin in the Wick rotated solution (30) prevent traversability.The behaviors of (34) and (30) are different at the origin except that, for boththe solutions, the throat appears at the same radius l0 = M = mβ. Similarconsiderations apply for the −ve branch.
Ellis III wormholes (34) can be straightforwardly extended to the rotatingform via the algotithm (11) and they would be likewise traversable, as has been
9
shown by Matos and Nunez [8]. Thus we do not deal with it here. Henceforth,we would rather concentrate on the Wick rotated Ellis I solution and ask: Canwe somehow remove the discontinuities in (30)? That is exactly where the newparamter a comes in. Let us see what happens when a 6= 0.
(b) a 6= 0 :The minimum area radius of the extended solution is obtained from the
equation dµ′
dl = 0 where ρ′(l) =√f ′−1(l2 +m2) is the area radius. From nu-
merical study of the resulting equation, we find that the minimum area occursat l < mβ and it decreases with the increase of a for fixed values of m and β.We also notice that the finite jump persists in the area radius of the extendedsolution f ′(l;m,β, a), δ being still given by Eq.(21). Surprisingly however, whena = 2mβ, the area function ρ′(l) decreases from +∞ to the minimum value atthe throat, then increases to a finite value at l = 0, undergoes no jump at l = 0but passes continuously, though not with C2 smoothness, across l = 0, on to−∞. The Ricci scalar R′ for f ′(l;m,β, a) is given by
R′ =8β(1 + β2
)(2mβ +
√4m2β2 − a2) exp[2β arccot( l
m )]
m(1 + l2
m2
)2[a2 +
(2mβ +
√4m2β2 − a2
)2
exp[4β arccot( lm )]
] (33)
and it approaches the value 4(1+β2)eπβ
m2(1+e2πβ)as l → ±0, that is, no jump in it.
The area behavior shows that, for the extreme case a = 2mβ, we do have atraversable wormhole with a single metric covering both the asymptotically flatuniverses (l → ±∞) connected by a finite wedge-like protrusion in the shapefunction at l = 0. This wedge prevents C2 continuity across l = 0 in the areafunction but sews up two exactly symmetrical asymptotically flat universes onboth sides. The numerical values of the free parameters m and β can alwaysbe suitably controlled to make the tidal force humanly tolerable and the travelsafer.
For a 6= 2mβ, such a single coordinate chart is not possible as the area hasa jump at l = 0. However, we can artificially circumvent this discontinuityand connect the two disjoint universes by multiple metric choices on differentsegments. We can get a cue for this construction from the static case. Considerthe metric form (18) on one segment (AB) and the Wick rotated metric (24) onthe other side (BC) so that the areas match at a radial point l = l1. The radiusl = l1 is a root of the equation (area from right (AB) = area from left (BC))
(m2 − l2) ×(l −m
l +m
)−β
= (m2 + l2) × Exp[2β arccot(l
m)] (34)
By numerical computation, it turns out that 0 < l1 < m such that the twootherwise disjoint universes, one represented by the branch AB and the otherbyBC, can be connected atB(l = l1). At the joining pointB, there is continuityin the area function (again not C2) and the tidal forces can be shown to be finitethroughout the generator curve ABC. Exactly similar arguments hold in therotating case. Branche AB belongs to the Wick rotated solution (f ′) while the
10
sector BC belongs to the original solution (f). Numerical calculations showthat the matching occurs at either of the two points B(l1) or B(l2) such that−m < l1, l2 < m.
Traversable wormholes can also be constructed by employing the “cut-and-paste” procedure [4]. One takes two copies of the static wormholes and joinsthem at a radius l = lb > l0. The interface between the two copies will thenbe described by a thin shell of exotic matter. The shape functions on bothsides will be symmetric. However, when rotation is introduced, numerical cal-culation shows that the throat radius decreases from the static value while theflaring out occurs faster. It is of some interest to note that Crisostomo and Olea[20,21] developed a Hamiltonian formalism to obtain the dynamics of a massiverotating thin shell in (2+1) dimensions. There, the matching conditions are un-derstood as continuity of the Hamiltonian functions for an ADM foliation of themetric. Of course, this procedure can be trivially extended to deal with axiallysymmetric solutions in (3+1) dimensions. For soliton solutions, see Ref.[22].
V. Extended Brans-Dicke I solution
To obtain the rotating Brans-Dicke solution, we pursue the following steps:Note from Eq.(3) that
√2ω + 3
2lnφ = ϕ = ln
[1 − m
2r
1 + m2r
]√2(β2−1)
⇒ φ =
[1 − m
2r
1 + m2r
]√4(β2−1)/(2ω+3)
.
(35)Now using the constraint from the Brans-Dicke field equations [9], viz.,
4(β2 − 1) = −(2ω + 3)C2
λ2 , (36)
where C, λ are two new arbitrary constants and ω is the coupling parameter,we get
φ =
[1 − m
2r
1 + m2r
]Cλ
=
[l −m
l +m
] C2λ
. (37)
The Eq.(34) can be rephrased in the familiar form [9]:
λ2 = (C + 1)2 − C(1 − ωC
2). (38)
However, the minimum area condition β2 > 1 requires that the right hand sideof Eq.(34) be positive. This is possible if either ω < − 3
2 or λ be imaginary. Letus first consider ω < − 3
2 so that the exponents are real. Then, the final step
consists in using the relation gµν = φ−1gµν together with replacing β in theexponents in the gµν by [7]
β =1
λ
(1 +
C
2
). (39)
11
This means we have the Brans-Dicke rotating class I solution for ω < − 32 as
follows:
ds2 = −f1(l)dt2 + f2(l)[dl2 + (l2 −m2)
(dθ2 + sin2 θdψ2
)], (40)
f1(l) ≡ f1(l;m,C, λ, a) = f(l;m,β, a)φ−1
=8mδ
[12λ(C + 2)
] [l−ml+m
]− 1λ (1+ C
2 )
a2 + 4δ2[
l−ml+m
]− 2λ (1+ C
2 )×[l −m
l +m
]− C2λ
, (41)
f2(l) ≡ f2(l;m,C, λ, a) = f−1(l;m,β, a)φ−1
=a2 + 4δ2
[l−ml+m
]− 2λ (1+ C
2 )
8mδ[
12λ(C + 2)
] [l−ml+m
]− 1λ (1+ C
2 )×[l −m
l +m
]− C2λ
, (42)
φ(l) =
[l −m
l +m
] C2λ
. (43)
It can be verified that the Brans-Dicke field equations again yield the expression
(36). Using the relation l = r + m2
4r , it can be easily expressed in the familiar(t, r, θ, ψ) coordinates with the value of δ given by Eq.(21) in which β should havethe value as in Eq.(36). For instance, when a = 0, we have δ = m
λ (C + 2) andidentifying m
2 = B, one retrieves the static Brans-Dicke metric in the originalnotation:
ds2 = −(
1 − Br
1 + Br
) 2λ
dt2 +
(1 +
B
r
)4(
1 − Br
1 + Br
) 2(λ−C−1)λ
×
[dr2 + r2dθ2 + r2 sin2 θdψ2] (44)
φ(r) =
(1 − B
r
1 + Br
)Cλ
. (45)
The solution (42) represents a naked singularity at r = B and the condition forthe existence of a minimum area is [6,7]
(C + 1)2 > λ2. (46)
For β2 > 1, and α = +1, the negative kinetic term in the field equations (5)shows that the energy density is negative violating at least the Weak EnergyCondition. The solution (38) still does not represent a traversable rotatingwormhole in the Einstein minimally coupled theory. One has to first make achange
m→ im, λ→ −iλ (47)
in (38) and (41) to get the Wick rotated counterpart. The next step is to usethe relations (32), (33) obtaining two branches as obtained in Sec.IV. Either
12
of the branches would then represent rotating traversable wormholes in theBrans-Dicke theory for the range of coupling values ω < − 3
2 . Same classes of
solutions will be obtained by alternative calculations with β2 > 1 and α = −1(imaginary scalar charge). The above steps represent the basic scheme to befollowed in other classes of solutions in the Einsten minimally coupled or Brans-Dicke theory, which we do not do here.
The discussion about traversability of the wormholes in the Einstein mini-mally coupled theory can be transferred almost in verbatim to the Brans-Dicketheory, once we use the crucial relation (37) connecting the parameters in boththe theories.
VI. Geodesic motion in the extended solution
We shall consider, as an illustration, the class of solutions (24) generatedfrom the Wick rotated seed solutions (18) of Einstein minimally coupled theory.The resulting solution is (now dropping primes)
f(l;m,β, a) =8mβδ exp[2β arccot( l
m )]
a2 + 4δ2 exp[4β arccot( lm)]
(48)
ϕ(l) =
[√2(1 + β2)
]arccot
(l
m
)(49)
so that the metric components gνλ are
g00 = −f, (50)
g11 = f−1, (51)
g22 = f−1(l2 + l20), (52)
g33 = f−1(l2 + l20) sin2 θ − fa2 cos2 θ, (53)
g03 = g30 = −fa cos θ. (54)
The four velocity is defined by
Uµ =dxµ
dp, xµ ≡ (t, l, θ, φ), dp = m0ds (55)
in which p is the new affine parameter and m0 is the invariant rest mass of thetest particle. The geodesic equations are given by
dUµ
dp− 1
2
∂gνλ
∂xµUνUλ = 0, gνλU
νUλ = m20 = ǫ (56)
Since the metric functions gνλ do not contain t and φ, the corresponding mo-menta are conserved, that is, the µ = 0 and µ = 3 equations give, respectively
U0 = −f.U0 − faU3 cos θ = k (57)
U3 = −faU0 cos θ + U3[f−1(l2 + l20) sin2 θ − fa2 cos2 θ] = h (58)
13
where k and h are arbitrary constants. From the above, it follows that the“angular momentum” of the particle is
r2(l)U3 =h− ka cos θ
sin2 θ≡ ξ (59)
r2(l) ≡ f−1(l2 + l20). (60)
The µ = 2 component of the geodesic equation gives
d
dp
(r2(l)
dθ
dp
)+
1
2faU0U3 sin θ − (U3)2[r2(l) + fa2] cos θ sin θ = 0. (61)
Instead of the µ = 1 equation, we take the second of Eq.(53) which gives
ǫ = −f [U0 + U3a cos θ]2 + f−1(U1)2 + r2(l)[(U2)2 + (U3)2 sin2 θ]. (62)
Looking at Eq.(58), we see that two solutions are possible. One is
U3 = 0 ⇒ θ = arccos(h
ka) = const.⇒ U2 = 0 (63)
which means θ can assume any constant value depending on the independentvalues of h, k and a. But such motions will only be radial since U2 = 0 andU3 = 0. In other words the gravitating source acts like a radial sink ! Thegeodesic equation of motion (59) reduces to:
(dl
dp
)2
= ǫf(l;m,β, a) + k2 (64)
This can be rewritten very succinctly as
d2l
dp2=ǫ
2
df
dl. (65)
The other solution of Eq.(58) is θ = 0, which implies that the test particlemotion is restricted to polar planes. However, we can always choose the poleperpendicular to this plane so that the angle ϕ varies in that plane with theparticle motion. Moreover, from Eq.(56), we must have h = ka so that r2(l)U3 =ξ 6= 0.Then, we end up again with the same metric function but without theexplicit appearance of a as an arbitrarily regulated free parameter. Thus, in theequation of motion, we have f = f(l;m,β, h/k) so that a, which is a parameterof the gravitating source, is obtained from the motional characteristics like hand k of the test particle itself. This intriguing feature is somewhat analogous tothe fact that the mass of the gravitating Sun can be determined from the motionof a test particle (planet) around it. As a special case, the parameter a can beso chosen as to completely mask the angular momentum ξ of the test particle,that is, as h → ka, there is a possibility that ξ → 0 too. Physically, it is likechoosing the parameter a to coincide with the orbital ϕ−angular momentum ofthe test particle. The equation of motion is again exactly the same as Eq.(59)
14
since U2 = 0 even though U3(6= 0) does not appear explicitly. This is due tothe fact that, in the last term of Eq.(58), (U3)2 sin2 θ = 0 but the signature oforbiting (non-radial) test particle is manifest in the presence of h:
(dl
dp
)2
= ǫf(l;m,β, h/k) + k2 (66)
The turning points of the orbit will occur when ǫf = −k2 and dfdl 6= 0. From
these conditions, we have the turning points occurring at
l = l0 = m cot
(lnx0
β
)(67)
where
x0 =−2mβǫ±
√4m2β2ǫ2 − k4a2
2k2δ. (68)
Circular orbits will occur if ǫf = −k2 and dfdl = 0 and they will be stable if
d2fdl2 < 0 and unstable if d2f
dl2 > 0.
VII. Sagnac effect
The best way to assess the effect of a non-zero U3 (or dϕ 6= 0) and of theMatos-Nunez parameter a is through the Sagnac effect [23] analyzing the orbitin the plane θ = 0. The effect stems from the basic physical fact that theround trip time of light around a closed contour, when the source is fixed ona turntable, depends on the angular velocity, say Ω, of the turntable. Usingspecial theory of relativity and assuming Ωr ≪ c, one obtains the proper timeδτs, when the two beams meet again at the starting point as
δτ s∼= 4Ω
c2S (69)
where S (≡ πr2) is the projected area of the contour perpendicular to the axisof rotation.
Without any loss of rigor, we take a → −a for notational convenience al-though it is not mandatory. Suppose that the source/receiver of two oppositelydirected light beams is moving along a circumference l = R = constant. Suit-ably placed mirrors reflect back to their origin both beams after a circular tripabout the central rotating wormhole. (The motion is thus not geodesic or forcefree!). Let us further assume that the source/receiver is moving with uniformorbital angular speed ω0 with respect to distant stars such that the rotationangle is
ϕ0 = ω0t. (70)
Under these conditions, the metric becomes
dτ2 = −f(R;m,β, a)[1 − aω0]2dt2 (71)
15
The trajectory of a light ray is dτ2 = 0 which gives
[1 − aω]2 = 0 (72)
where ω is the angular speed and where we have suspended the condition h = kafor photon orbits. The roots of the above equation coincide and is
ω1± =1
a. (73)
Therefore, the rotation angle for the light rays is
ϕ = ω1±t = ω1±ϕ0
ω0. (74)
The first intersection of the world lines of the two light rays with the world lineof the orbiting observer after emission at time t = 0 occurs when
ϕ+ = ϕ0 + 2π, ϕ− = ϕ0 − 2π (75)
so thatϕ0
ω0ω1± = ϕ0 ± 2π (76)
where ± refer to co-rotating and counter-rotating beams respectively. Solvingfor ϕ0, we get
ϕ0± = ± 2πω0
ω1± − ω0. (77)
The proper time as measured by the orbiting observer is found from Eq.(66) byusing dt = dϕ0/ω0 and integrating between ϕ0+ and ϕ0−. The final result isthe Sagnac delay given by
|δτ s| =√f(R;m,β, a)
[1 − aω0
ω0
](ϕ0+ − ϕ0−)
=√f(R;m,β, a)
[1 − aω0
ω0
] [4πω0
1a − ω0
].
=√f(R;m,β, a)(4πa). (78)
This shows remarkably that the Sagnac delay depends only on the Matos-Nunezparameter a. Interestingly, the result is independent of ω0 meaning that it isindependent of the motional state of the source/receiver, be it static with respectto the frame of distant stars or moving with regard to it. When a = 0, there isno delay because the wormhole spacetime is then nonrotating (no turntable!).The above result supports the conclusion of Ref.[8] from an altogether differentviewpoint that a can indeed be interpreted as a rotational parameter of thewormhole.
VIII. Summary
16
Asymptotically flat rotating solutions are rather rare in the literature, bethey wormholes or naked singularities. The algorithm (11), together with someoperations, provides a method to generate new traversable wormhole solutionsin the Einstein minimally coupled and then in the Brans-Dicke theory. Thepresent study opens up possibilities to explore in more detail new solutions inother theories too. For instance, the string solutions are just the Brans-Dickesolutions with ω = −1. As we saw, the asymptotically flat wormhole solutionsadmit two arbitrary parameters a and δ. The Matos-Nunez parameter a hasbeen interpreted by the authors (Ref [8]) as a rotation parameter. Here we haveshown how the parameters can be adjusted for obtaining traversability.
The static Ellis I wormholes do not appear traversable due to the singularityas manifested in the behavior of the area function and curvature. To tacklethis problem, we analytically continued it via Wick rotation and rederived El-lis III singularity free, asymptotically flat, traversable wormholes as one of itsbranches. This showed that the two classes of solutions are not independent,contrary to the general belief. The comparative features of the Wick rotatedand Ellis III solutions are pointed out. In the extended Wick rotated solution,numerical graphics showed that the wormhole can be covered by a single metricwith C0 continuity. That is, the jumps can be sewed up at the origin for theextreme value of a (= 2mβ) or can be avioded by choosing multiple metricpatches for nonextreme values of a, the junctions again having only C0 conti-nuity. In either case, the tidal forces depend on controllable parameters m andβ so that practical traversability is assured. As an illustrative seed solution,we considered only the Ellis I solution in the Einstein minimally coupled scalarfield theory and finally mapped the extended solution into that of Brans-Dicketheory.
In connection with the static Brans-Dicke I solution, we recall a long stand-ing query by Visser and Hochberg [24] which has been the guiding motivation inSecs. II-V: “It would be interesting to know a little bit more about what this re-gion actually looks like, and to develop a better understanding of the physics onthe other side [that is, across the naked singularity] of this class of Brans-Dickewormholes.” The analyses above answer how one could achieve a both sidedasymptotically flat traversable singularity free wormhole via Wick rotation ofthe Ellis I solution which is merely the conformally rescaled Brans-Dicke I solu-tion. Because of the fact that Ellis III is rederivable from Ellis I and that theformer is a traversable wormhole, we can say that Brans-Dicke I solution is alsoa traversable wormhole, but only in its Ellis III reincarnation. We believe thatthis argument provides some understanding of the “other side”: The mathemat-ical operations of conformal rescaling plus Wick rotation plus a trigonometricrelation eases out the naked singularity and converts the Brans-Dicke I solutioninto a traversable wormhole.
The next task was to investigate into the extended solutions with new pa-rameters. Thus, in section VI, we studied the geodesic motion in the extendedgeometry and obtained remarkable new results in the Einstein minimally cou-pled theory: For nonzero values of constant θ, the spacetime acts like a radialsink. For θ = 0, the spacetime allows nonradial motions (U3 6= 0) but the
17
Matos-Nunez parameter a can be entirely expressed in terms of the constants ofmotion. In Sec. VII, we calculated the Sagnac effect in the extended spacetimeand found that the delay depends on a. If a = 0, the delay is zero imply-ing that a could be interpreted as a rotation parameter thereby supporting theconclusion of Matos and Nunez [8] from an altogether different viewpoint.
A very pertinent question is the stability of wormholes in the scalar-tensortheory, especially in the Einstein minimally coupled theory under considerationhere. To our knowledge, the instability of black hole solutions in the Einsteinconformally coupled theory was shown first by Bronnikov and Kireyev [23]. Asto the wormhole solutions, it has been shown by Bronnikov and Grinyok [17] thatthey are also unstable under spherical perturbations δϕ of the scalar field satis-fying ϕ:µ
;µ = 0. Instabilities occur due to the occurrence of negative energy levelsin the effective potential. The authors suspected that such instabilities couldbe generic features of scalar-tensor theories. Instability of massive wormholesin the Einstein minimally coupled theory has been shown by Armendariz-Pıcon[17]. However, he also showed that small mass wormholes in that theory couldstill be stable. It should be of interest to study whether the extended solutionsare stable under perturbations. This remains a task for the future.
Acknowledgments:
The authors are deeply indebted to Guzel N. Kutdusova of BSPU for efficientand useful assistance. We thank three anonymous referees for useful criticismsand suggestions.
References
[1] M.S. Morris and K.S. Thorne, Am. J. Phys. 56, 395 (1988); M.S. Morris,K.S. Thorne and U. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988).
[2] A. Einstein and N. Rosen, Phys. Rev. 48, 73 (1935).[3] D. Hochberg, Phys. Lett. B 251, 349 (1990).[4] M. Visser, Lorentzian Wormholes- From Einstein to Hawking (A.I.P.,
New York, 1995).[5] A.G. Agnese and M. La Camera, Phys. Rev. D 51, 2011 (1995); K.K.
Nandi, A. Islam and J. Evans, Phys. Rev. D 55, 2497 (1997).[6] C. Barcelo and M. Visser, Class. Quant. Grav. 17, 3843 (2000); B.
McInnes, J. High Energy Phys. 12, 053 (1002); G. Klinkhammer, Phys. Rev.D 43, 2512 (1991); L. Ford and T.A. Roman, Phys. Rev. D 46, 1328 (1992);ibid, 48, 776 (1993); L.A. Wu, H.J. Kimble, J.L. Hall and H. Wu, Phys. Rev.Lett. 57, 2520 (1986).
[7] K.K. Nandi, B. Bhattacharjee, S.M.K. Alam and J. Evans, Phys. Rev.D 57, 823 (1998); L. Anchordoqui, S.P. Bergliaffa and D.F. Torres, Phys. Rev.D 55, 5226 (1997).
[8] T. Matos and D. Nunez, Class. Quant. Grav. 23, 4485 (2006) andreferences therein; T. Matos, Gen. Rel. Grav. 19, 481 (1987).
[9] C.H. Brans, Phys. Rev. 125, 2194 (1962).[10] A. Bhadra and K.K. Nandi, Mod. Phys. Lett. A 16, 2079 (2001).[11] A. Bhadra and K. Sarkar, Gen. Rel. Grav. 37, 2189 (2005).[12] S. Kar, Class. Quant. Grav. 16, 101 (1999)[13] Y.M. Cho, Phys. Rev. Lett. 68, 3133 (1992).
18
[14] G. Clement, Phys. Lett. B 367, 70 (1996); Grav. & Cosmol. 5, 281(1999).
[15] C. Martinez, C. Teitelboim and J. Zanelli, Phys. Rev. D 61, 104013(2000).
[16] H.A. Buchdahl, Phys. Rev. 115, 1325 (1959).[17] C. Armendariz-Pıcon, Phys. Rev. D 65, 104010 (2002).[18] K.K. Nandi, Y.-Z. Zhang, Phys. Rev. D 70, 044040 (2004); K.K. Nandi,
Y.-Z. Zhang and K.B. Vijaya Kumar, Phys. Rev. D 70, 127503 (2004).[19] H.G. Ellis, J. Math. Phys. 14, 104 (1973), ibid, 15, 520E (1974).[20] J. Crisostomo and R. Olea, Phys. Rev. D 69, 104023 (2004).[21] R. Olea, Mod. Phys. Lett. A 20, 2649 (2005).[22] O. Bertolami, Phys. Lett. B 234, 258 (1990).[23] A. Tartaglia, Phys. Rev. D 58, 064009 (1998); K.K. Nandi, P.M. Alsing,
J.C. Evans and T.B. Nayak, Phys. Rev. D 63, 084027 (2001); A. Bhadra, T.B.Nayak and K.K. Nandi, Phys. Lett. A295, 1 (2002).
[24] M. Visser and D. Hochberg, in Proceedings of the Haifa Workshop on“The internal structure of black holes and space time singularities”, Jerusalem,Israel, June, 1997, [gr-qc/970001], p.20.
[16] The solutions have been rediscovered repeatedly after they were first de-rived by I.Z. Fisher, Zh. Eksp. Teor. Fiz. 18, 636 (1948).
[17] The present authors have found the above earliest reference from K.A.Bronnikov and S. Grinyok, Grav. & Cosmol. 7, 297 (2001).
[18] H.A. Buchdahl[19] K.A. Bronnikov, Acta Phys. Polon. B4. 251 (1973). The form of
the solution given herein in “harmonic” coordinates ( t, u, θ, ϕ) can be easilytransferred to the Janis-Newman-Winnicour “standard” form ( t, ρ, θ, ϕ) whichis equivalent to the Buchdahl “isotropic” form. See, respectively, Refs.[17] andA.Bhadra and K.K.Nandi, Int.J.Mod.Phys. A16, 4543 (2001); Janis-Newman-Winnicour, PRL
[20] K.A. Bronnikov and G.N. Shikin, Grav. & Cosmol. 8, 107 (2002).[21] We thank an anonymous referee for pointing this out.[22] The factor
√2 appears because we took α = +1 instead of +2. Other
identifications of the parameters with the original solution in [19] are: (m, a, n)
of Ellis are our (M,m,m√
1 + β2).[23] K.A. Bronnikov and Yu. N. Kireyev, Phys. Lett. A67, 95 (1978).
Appendix
A wormhole is defined to be traversable by a hypothetical traveler if it satis-fies some general constraints [1]. We shall demonstrate that the Ellis III worm-hole satisfies all of these. It can be made traversable even by a human travelerunder suitable choices of constants m and β. Let us put one branch, say, the+ve branch of Eq.(29) into the metric (24) and rewrite it in the standard MTYform [1] by defining a radial variable ρ as
(l2 +m2) exp[2βπ2− arctan(
l
m)] = ρ2. (A1)
19
(Note that l → ±∞ implies ρ → ±∞ and l → ±0 implies ρ → ±meπβ .) Thenthe metric (24) in the coordinates (t, ρ, θ, ψ) becomes
ds2 = −e2Φ(ρ)dt2 +dρ2
1 − b(ρ)ρ
+ ρ2(dθ2 + sin2 θdψ2) (A2)
where the redshift function Φ is
Φ(ρ) = β
[arctan
l(ρ)
m
− π
2
](A3)
and the shape function b is
b(ρ) = ρ
[1 − [l(ρ) −mβ]2
ρ2exp[2βπ
2− arctan(
l(ρ)
m)]]. (A4)
General constraints on b and Φ to produce a traversable wormhole are satifiedby the functions in (A3) and (A4). It may be verified that: (1) Throughout
the spacetime, 1 − b(ρ)ρ ≥ 0 and b(ρ)
ρ → 0 as ρ → ±∞ (2) Throat occurs at the
minimum of ρ where b(ρ) = ρ. This minimum ρ0 corresponds to l0 = mβ sothat
ρ0 = m(1 + β2)12 exp[βπ
2− arctanβ]. (A5)
(3) The spacetime (A2) has no horizon, that is, Φ is everywhere finite. (4) Thecoordinate time t measures proper time in asymptotically flat regions becauseΦ → 0 as ρ → ±∞. (5) The spacetime has no singularities, as discussed inSec.IV.
Some more constraints are necessary if the trip is to be undertaken by ahuman traveler: (a) Trip begins and ends at stations located on either side ofthe throat where the gravity field should be weak. This demands that (i) the
geometry at stations must be nearly flat, or, b(ρ)ρ ≪ 1, (ii) the gravitational
redshift of signals sent from stations to infinity must be small, or, |Φ| ≪ 1and (iii) the acceleration of gravity at the stations must be less than one Earth
gravity g⊕ = 980cm sec−2, or,∣∣∣c2(1 − b
ρ)12
dΦdρ
∣∣∣ . g⊕. While the first two con-
straints (i) and (ii) are easily met in virtue of the general constraints (1) and(4) respectively, (iii) gives
∣∣∣∣mβ
l2 +m2eΦ∣∣∣∣ . .
g⊕c2. (A6)
For fixed finite values of m and β, eΦ → 1 for large l and hence this constraintcan be easily satisfied at the stations. (b) The tidal forces suffered by thehuman traveler should be tolerable which means that the magnitude of thedifferential of four acceletation |∆−→a | should be less than g⊕ in the orthonormalframe (eb0′ , eb1′ , eb2′eb3′) of the traveler. This constraint translates, for a traveler of
20
length (head to foot) ∼ 2meters, into the following bounds on the componentsof curvature tensor computed in his/her frame:
∣∣Rb1′ b0′ b1′ b0′
∣∣ =
∣∣∣∣∣
(1 − b
ρ
)(−d
2Φ
dρ2+
ρ dbdρ − b
2ρ(ρ− b)
dΦ
dρ−(dΦ
dρ
)2)∣∣∣∣∣ (A7)
/g⊕
c2 × 2metr.≃ 1
(1010cm)2.
This bound is meant to constrain Φ(ρ) which is already well behaved at thethroat ρ = ρ0 where its value is β
[arctanβ − π
2
]. This value is −1 as β → ∞,
and 0 as β → 0. In general, as ρ → ±∞, Φ vanishes. Thus the constraint(A7) is easily satisfied. This result is expected since all the curvature tensorcomponents fall off with distance and vanish at infinity [19]. Let us look at thelateral bounds given by
∣∣Rb2′ b0′ b2′ b0′
∣∣ =∣∣R
b3′ b0′ b3′ b0′
∣∣ =∣∣∣∣γ2
2ρ2
[(vc
)2(db
dρ− b
ρ
)+ 2(ρ− b)
dΦ
dρ
]∣∣∣∣ (A8)
/g⊕
c2 × 2metr.≃ 1
(1010cm)2
which constrain the speed v with which the traveler crosses the wormhole, γ =(1 − v2/c2)−
12 . For the metric (A2), the above works out to
∣∣Rb2′ b0′ b2′ b0′
∣∣ =∣∣R
b3′ b0′ b3′ b0′
∣∣ =∣∣∣∣γ2
2ρ2
[(vc
)2(
2m(m+ lβ)
l2 +m2
)+
(2mβ(l −mβ)
l2 +m2
)]∣∣∣∣(A9)
/g⊕
c2 × 2metr.≃ 1
(1010cm)2.
The maximum tidal force should be experienced by the traveler at the throatat l = mβ or equivalently at ρ = ρ0. Hence the velocity v at the throat isdetermined by the following constraint
γ2
ρ20
[(vc
)2]
.1
(1010cm)2. (A10)
It is possible to adjust m and β such that we have a throat radius ρ0 ∼ 10metr.(say). Assuming that v ≪ c or γ ∼ 1, we obtain v . 30metr./s( ρ0
10metr. ) whichshows that the speed v across the hole could be made reasonably small. (c) Thetraveler should feel less than g⊕ acceleration throughout the trip which requires
∣∣∣∣∣e−Φ
(1 − b
ρ
) 12 d
dρ
(γeΦ
)∣∣∣∣∣ .
g⊕c2. (A11)
With γ ∼ 1, this works out to the same constraint as in (A6) which is alreadysatisfied. (d) The total proper time interval ∆τ measured by the traveler and
21
the coordinate time interval ∆t measured at the stations for the entire tripshould be about a year (say), that is,
∆τ =
∫ L2
−L1
dL
vγ. 1year (A12)
∆t =
∫ L2
−L1
dL
veΦ. 1year (A13)
where L = −L1 and L = +L2 are the proper distances of the stations measured
from the throat. Since γ ∼ 1, ∆τ ≃ (L2 +L1)/−v where
−v is the average velocity
not exceeding 30metr./s for a 10 metr. throat radius. This suggests that thetotal separation between the stations should be of the order of 9.4 × 105Km.
It is expected that the rotating wormhole should also be traversable, theonly physical difference is that the travelers would feel an additional centrifugalforce. However, the mathematical treatment of traversability criteria in therotating case is quite cumbersome. We leave it as a future task.
Consider the rotating solution, metric (7), with f generated from the +vebranch of Ellis III solution (29) given by
f(l;m,β, a) =8mβδ exp[2β+π
2 − arctan( lm )]
a2 + 4δ2 exp[4β+π2 − arctan( l
m )].
Since the metric components in (7) do not depend on t, the vector ∂∂t is a Killing
vector of the spacetime but is not everywhere orthogonal to the constant t−hypersurfaces. Let us use a time variable T by
dT = dt+ a cos θdψ.
The vector ∂∂T is not a Killing vector but is everywhere orthogonal to the con-
stant T−hypersurfaces. Hence the metric (7) becomes
ds2 = −fdT 2 + f−1[dl2 + (l2 + l20)(dθ
2 + sin2 θdψ2)].
Under the redefinition dT = dt′ +√
1−ff dl, the metric (A) becomes
ds2 = −dt′2 + (dl −√
1 − fdt′)2 + (l2 + l20)(dθ2 + sin2 θdψ2).
It was shown in Eq.(63) of Sec.VI that one solution of the geodesic equationswas radial motion (U3 = 0) on the plane θ = const. Without loss of generalitywe can choose θ = π/2 which implies that dT = dt on the equatorial plane. Themetric on the constant t′ surface for θ = π/2 is then dσ2 = dl2 + (l2 + l20)dψ
2.
This surface is isometric to a catenoid (x, y, z) | (x2 + y2)12 = l0 cosh( z
l0) in
E3, the radius of the central hole being l0 [19]. The catenoid with a central holeis the sink referred to in Sec.VI. The generators of the catenoid are the radialtrajectories of a hypothetical traveler.
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